   Definition 2.8 [1]}
Let  \(\left\lbrace \,f_{\,i}\,\right\rbrace _{i \,=\, 1}^{\infty }\)  be a frame for  \(H\) .Then the bounded linear operator  \( T \,:\, l^{\,2}\,(\,\mathbb {N}\,) \,\rightarrow \, H\) , defined by  \(T\,\lbrace \,c_{i}\,\rbrace  \,=\, \sum \limits _{i \,=\, 1}^{\infty } \;c_{\,i}\,f_{\,i}\) , is called pre-frame operator and its adjoint operator  \(T\,^{\,\ast } \,:\, H \,\rightarrow \, l^{\,2}\,(\,\mathbb {N}\,)\) , given by \(T^{\,\ast }\,f \,=\, \left\lbrace \, \left<\,f \,,\, f_{i} \,\right>\,\right\rbrace _{i \,=\, 1}^{\infty }\)  is called the analysis operator.The operator  \(S \,:\, H \,\rightarrow \, H\)  defined by  \(S\,f \,=\, T\,T^{\,\ast }\,f \,=\, \sum \limits ^{\infty }_{i \,=\, 1}\; \left<\,f \,,\, f_{\,i} \, \right>\,f_{\,i}\)  is called the frame operator.

